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To drag

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Jingxuan Zhang:
I am referring to the mean value theorem proof in
http://www.math.toronto.edu/courses/apm346h1/20181/PDE-textbook/Chapter7/S7.2.html

So twice we dragged the kernel $G$ out of the integral. How can we actually do that? if $\Delta u \gtrless$ can we still drag? can we in general drag?

Victor Ivrii:
We drag out of integral only what does not depend on the variable of integration, so from the point of view of integral it is a constant.

There is a more risky trick: changing the order of integration; it needs to be justified since the integrand are singular––but for the full rigour you need to go to MAT351 or even graduate course.

Jingxuan Zhang:
But I think it's quite apparent that $G(x,y)$ depends on the variable of integration, both $dV$ and $dS$?

Victor Ivrii:

--- Quote from: Jingxuan Zhang on March 13, 2018, 10:08:27 PM ---But I think it's quite apparent that $G(x,y)$ depends on the variable of integration, both $dV$ and $dS$?

--- End quote ---
Observe that we drag it out from the integral over the sphere with radius $\varepsilon$ centered at $y$; and there $G(x,y)$ and $\frac{\partial G}{\partial \nu_x}$ are constant